The fractal C is constructed as follows. (i) Begin with the line segment Io = [0, 1]. (ii) Remove the open middle third (§,), to get I1 = [0, U, 1]. (iii) Remove the open middle third from every remaining line segment, to get [2 7 8 U 3'9 12 = 0, U (iv) Repeat the process of removing the open middle thirds of the remaining line segments infinitely many times. The first few iterations of this construction are illustrated below. (a) Find the total length of the intervals removed from [0, 1] after three iterations, i.e. the total length of the intervals removed from [0, 1] to get I3. (b) By considering a suitable series, show that the total length of all of the intervals removed is equal to 1. (c) Despite the fact that an infinite number of intervals of total length 1 have been removed to construct C, there are points in the interval [0, 1] that will never be removed, i.e. there are points in C. What is one such point?

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Suppose we used our failing perfect shuffling of digits to mix the digits of the numbers (x,y) that describe a point on the square to get a one-to-one correspondence with the points on a line segment. What point in the square would be paired with the point 0.12000100010001000100010001...?

(c) √ z²dx-zdy+2ydz; C: the line segments from (0,0,0) to (0,1,1), from (0,1,1) to 77 (1,2,3), and from (1,2,3) to (1,2,4). (ans: 6